# Trig Heuristic Method - Alpha V1.0 - C++ version - 11/18/15

`//Project starts here: special case when objective variable distance to the origin intersects the k-th constraints is not yet address here`

`//Two variable optimization project, which includes more than three constrains`

`#include <iostream> // std::cout, std::cin`

`#include <cmath> // std:: sqrt, std::pow, std::abs`

`using namespace std;`

`int i=0, j=0; // incremental operators`

`int a=0; // incremental operator in an array that is used to record the smallest angle for the shortest distance gamma, among equally small angles`

`int b=0; // incremental operator used to organize constraints bound by the k-th constraint; these constraints are below a threshold that requires them to considered in the set of primary constraints`

`int f=0; // incremental operators`

`int r=0; // operator set to sum the number of primary reference constraints, which excludes erased secondary set of reference constraints, and system constraints, which are established based on the k-th bounds`

`int q=0; // operator used to identify second reference constraint after the k-th one; this constraint is included in a primary set of constraint that are not bounded by the k-th constraint`

`int k=0; // operator used to identify first reference constraint, which yields the shortest distance gamma and smallest angle.`

`int kk=0; //resequenced k-th constraint`

`long double p=0; // operator used to subtract one array from another, and operator used towards computing the power function`

`int num=2; // number of variables in the objective function`

`int t=6; // number of constraints`

`int u=0; // increment operator used to assign value to array beyond the 6th referenced constraint`

`int g=0; // operator used to replace the primary set of reference constraints and erase secondary set of constraints`

`long double newconstr[8][2]; // matrix containing six of the original constraints and two project generated constraints`

`long double formatconstr[8][2]; // matrix containing set of primary reference constraints, which excludes erased secondary set of reference constraints`

` `

`long double epi[2][2]; // matrix inverse of the matrix containing two constraints meant to be used for obtaining the quantity for each variable`

`long double x=0; // operator used to sum the squares, to measure distance between constraint coordinates and variable coefficients, by taking the square root for the sum of squares`

`int y=0; // identifier when the angle for the k-th constraint is null`

`int s=0; // identifier for goto loop, when angle for th k-th constraint is null`

`long double z=0; // operator used to sum the squares, to measure distance between variable coefficients and origin`

`int LC[8] = {}; `

`long double LK[8] = {}; // measure distance between the k-th constraint and second likely constraint`

`long double L7[8] = {}; // array used to store angles`

`long double LA[8] = {}; // measure distance between constraint coordinates and origin; there are six constraints specified by the user and two system generated`

`// array used to store the distance between system constraint coordinates and origin; there are six constraints specified by the user and two system generated`

`long double L6[8] = {}; // measure distance between constraint coordinates and variable coefficients; there are six constraints specified by the user and two system generated`

`// array used to store the distance between system constraint coordinates and variable coefficients; there are six constraints specified by the user and two system generated`

`long double L8[8] = {}; // compute cosine, and thereafter record the angle, value of the triangle formed between the variable coefficients, a given constraint, and the origin`

`long double L9[8] = {}; // multiply distance between the variable coefficients and a given constraints and the cosine value; this provides distance between the two points if they were aligned between the variable coefficients and the origin`

`long double LB[8] = {}; // array that includes all constraints that will be excluded from the set of primary constraints, since the array includes all constraints bound by the k-th constraint below a target threshold`

`long double LE[8] = {}; // array used record the angle of the constraint yielding the smallest distance`

`int LF[8] = {}; // array used store which constraint yields the smallest distance`

`// array used to store the distance between system constraint coordinates and k-th constraint`

`long double w=0, v=0; // angles recorded from triangles formed, which are used to determine which plane formed by the binding a system constraint and the k-th constraint intersects the line formed by binding the coefficient variables and the origin`

`long double sol=0; // operator used to sum parameters for each constraint; this is equivalent to a sumproduct of the constraint with a unity vector`

`long double SOLL[2]; // array used to store the solution set for the maximization project`

`long double N[2] = { 18.3333333333333333, 20 }; // array of coefficients for variables in the objective function, which will be used in the maximization project within the prescribed constraint`

`// parameters for each of the six constraints are assigned to the variable to be optimized`

`long double constraints[6][2]= {`

`{1, 2}, `

`{3, 4},`

`{5, 6}, `

`{15, 12}, `

`{12, 15}, `

`{11, 12}`

`};`

`// limiting parameters for each of the six constraints`

`long double limit[6]= {`

`10, 20, 30, 70, 70, 60`

`};`

`//list of functions`

`void cosine_and_angle(long double array1[8], long double array2[8], int upper_bound); //compute cosine value and angle of the triangle formed between the variable coefficients, a given constraint, and the origin`

`void min(long double array3[8], int upper_bound); //function used to determine the minimum within an array and below an upper bound`

`void matrix_inverse(long double array4[2][2], int amount6, int amoutn7); //function used to compute the inverse of a 2 x 2 matrix`

`long double cosine(long double amount1, long double amount2, long double amount3); //function used to compute the cosine `

`void dist_constr_var (long double array5[8][2], int amount5, int amount4, long double return_array[8], int upper_bound, int increment); //measure distance between constraint coordinates and variable coefficients, by taking the square root for the sum of squares`

`//int main begins`

`int main () `

`{`

` `

` // defined coefficients are provided for each variable of the objective function for the maximization problem`

` for(i=0;i<t;i++){`

` for(j=0;j<num;j++){`

` newconstr[i][j] = constraints[i][j]/ limit[i];`

` }`

` }`

` `

` //measure distance between constraint coordinates and origin, by taking the square root for the sum of squares`

` dist_constr_var (newconstr, 0, 0, LA, t, 0) ;`

` `

` //measure distance between constraint coordinates and variable coefficients, by taking the square root for the sum of squares`

` dist_constr_var (newconstr, 0, 1, L6, t, 0) ;`

` //measure distance between variable coefficients and origin, by taking the square root for the sum of squares`

` for(j=0;j<num;j++){`

` z += pow(N[j],2);`

` }`

` z = sqrt(z); //measure distance between variable coefficients and origin`

` `

` //compute cosine value and angle of the triangle formed between the variable coefficients, a given constraint, and the origin `

` cosine_and_angle(LA,L6,t) ; `

` `

` //find the smallest distance gamma recorded in array L9`

` min(L9, t);`

` //record all constraints for which the smallest distance gamma is equal`

` for(i=0;i<t;i++){`

` if (L9[i] == x) {`

` LF[a]=i; //array used record which constraint yields the smallest distance gamma`

` LE[a]=L8[i]; // array used record the angle of the constraint yielding the smallest distance gamma`

` a++;`

` }`

` }`

` //record the smallest angle for the shortest distance gamma, which is recorded in the set of ‘’a’’ for equally small angles`

` min(LE, a);`

` `

` //record the k-th constraint that yields the shortest distance gamma and angle, which is the first reference constraint`

` for(i=0;i<a;i++){`

` if (LE[i] == x) {`

` k = LF[i];`

` x=0; `

` }`

` }`

` `

` //determnine if the anlge for the k-th constraint is null`

` for(j=0;j<num;j++){`

` LE[j] = N[j]/ newconstr[k][j];`

` x += LE[j];`

` }`

` `

` for(j=0;j<num;j++){`

` if ( x/num - LE[j] < 1e-10) { // y==1, this case applies when the angle for the k-th constraint is null, within rounding erro `

` p++;`

` }`

` }`

` `

` if (p == num){`

` y = 1;`

` }`

` `

` p = 0;`

` `

` //mapping of back-up constraints by using the k-th constraint as binding`

` newconstr[6][0] = newconstr[k][0];`

` newconstr[6][1] = 0;`

` newconstr[7][0] = 0;`

` newconstr[7][1] = newconstr[k][1];`

` if (y != 1) {`

` //record the distance between the newly created system constraint, by taking the square root of the sum of squares`

` dist_constr_var (newconstr, 0, 0, LA, num, t) ;`

` dist_constr_var (newconstr, 0, 1, L6, num, t) ;`

` //measure distance between k-th constraint and system generated constraint`

` dist_constr_var (newconstr, 1, 0, LK, num, t);`

` //back-up constraints are now available`

` //measure the angle for each triangle and verify whether the angle formed by the k-th constraint is the difference between the former two`

` for(i=0;i<num;i++){`

` u = i+t;`

` v = cosine(LK[u], L6[u], L6[k]);`

` v = acos(v);`

` `

` w = cosine(LA[u], z, L6[u]);`

` w = acos(w);`

` //determine which of the two system constraints should be used as the second reference L-th constraint with the k-th one`

` `

` //computational error prevents the certitude of additional for angles; an error of 1e-17 is chosen as an acceptable threshold`

` if (abs( L8[k] + w - v )< 1e-10 ){`

` q=i+t;`

` }`

` }`

` for(j=0;j<num;j++){`

` if (newconstr[q][j] == newconstr[k][j]){`

` a = j;`

` v = newconstr[k][j];`

` }`

` }`

` }`

` kk=k;`

` repeat:`

` k=kk;`

` //verify whether all constraints, which exclude the k-th constraint, are bound by the k-th primary constraint and its shared axis with the q-th constraint`

` //repeat:`

` b=0;`

` g=0;`

` `

` u = 2+t;`

` for(i=0;i<u;i++) {`

` f=0;`

` x=0; `

`// if and only if the angle for the k-th constraint is not null, then all constraint that are below a threshold, which is defined by the segment binding the k-th constraint and the q-th constraint, will chosen for removal`

` `

`// review all constraints not excluded from the set of primary constraints, since the array includes all constraints bound by the k-th constraint below a target threshold`

` if (y != 1) {`

` if (i != k) { `

` if (newconstr[i][a] < v) {`

` f++;`

` }`

` if (newconstr[i][a] >= v) {`

` x++;`

` p++;`

` }`

` }`

` if (i == k) {`

` x++;`

` }`

` }`

` if (y == 1) {`

` if (i != k) {`

` if (newconstr[i][s] < newconstr[k][s]) {//repeat loop for each "s" coordinate of the k-th constraint`

` f++;`

` }`

` `

` if (newconstr[i][s] >= newconstr[k][s]) {//repeat loop for each "s" coordinate of the k-th constraint`

` x++;`

` p++;`

` }`

` }`

` if (i == k) {`

` x++;`

` }`

` }`

` if ( f != 0 ) {`

` LB[b] = i; //i-th constraint will be recorded in the LB array, which includes all constraints that will be excluded from the set of primary constraints`

` b++;// ”b” operator indicates the total number of constraints bound by the k-th constraint`

` }`

` if ( x != 0 ) {// array used to sort the reference set of primary constraints`

` LF[g] = i;`

` if (i == k) {`

` k=g;`

` }`

` g++;`

` }`

` }`

` f=0;`

` // if the number of constraints excluded from the set of primary constraints amounts to the total number constraints minus one, which is the k-th constraint, then the k-th constraint is the only binding one`

` a=0;`

` g=0;`

` `

` if (p == 0) { a=1; } // a is an operator used to determine whether the first reference constraint is the only binding constraint`

` p=0;`

` `

` if (a == 1) {`

` `

` //first constraint reference provides bounds for solution; the L-th and k-th constraints are used to determine the solution`

` //the following provide the inverse of a 2x2 matrix and output of solution`

` if (y != 1) {`

` matrix_inverse(newconstr, q, k);`

` }`

` `

` if (y == 1) {`

` for(i=0;i<num;i++){`

` q=i+t;`

` matrix_inverse(newconstr, q, k);`

` cout<< "\n";`

` }`

` }`

` `

` s++;`

` if (s < num) {`

` goto repeat;`

` }`

` `

` return 0; //the project ends here, if a=1`

` }`

` `

` //matrix containing set of primary reference constraints, which excludes erased secondary set of reference constraints`

` r = (u-b);`

` for (i=0;i<r;i++){`

` a=LF[i];`

`LE[i]=LF[i];`

` for (j=0;j<num;j++){`

` formatconstr[i][j] = newconstr[a][j];`

` }`

` }`

` `

` //measure distance between constraint coordinates and origin, by taking the square root for the sum of squares`

` dist_constr_var (formatconstr, 0, 0, LA, r, 0) ;`

` //measure distance between constraint coordinates and variable coefficients, by taking the square root for the sum of squares`

` dist_constr_var (formatconstr, 0, 1, L6, r, 0) ;`

` //recalculating cosine values`

` cosine_and_angle(LA,L6,r) ; `

` //begin to locate second reference constraint`

` dist_constr_var (formatconstr, 1, 0, LK, r, 0);`

` `

` f=0;`

` for(i=0;i<r;i++){`

` if (i != k) {`

` // angle recorded from the reference k-th constraint and likely second constraint`

` v = cosine(LK[i], L6[k], L6[i]);`

` v = acos(v); // determine angle v `

` `

` //sum of angles should amount to the addition of the k-th constraint angle and likely second reference constraint;`

` //this provides an array of distances between the k-th constraints and other constraints; these distances must intersect with the distance from the origin to the objective variable coordinates`

` //computational error prevents the certitude of additional for angles; an error of 1e-12 is chosen as an acceptable threshold rather than 1e-17`

` `

` // L6 array used to store the second reference constraint after the k-th one; this constraint is included in a primary set of constraints that are not bounded by the k-th constraint`

` `

` if (abs( L8[k] + L8[i] - v ) < 1e-10) {`

` L6[f] = i;`

` f++;`

` }`

` }`

` }`

` `

` //determine angle formed between a slope orthogonal to gamma and the distance between the k-th constraint and the likely second reference constraint`

` for(i=0;i<f;i++){`

` a = L6[i];`

` L7[i] = acos( cosine( L9[a], L9[k], LK[a] ) );`

` }`

` `

` //determine the minimum`

` min(L7, f);`

` for(i=0;i<f;i++){`

` if (L7[i]==x){`

` q=L6[i];`

` a=q;`

` }`

` }`

` `

` if (y != 1) {`

` matrix_inverse(formatconstr, q, k); //the following provide the inverse of a 2x2 matrix and output solution `

` x=0;`

` p=0;`

` for(i=0;i<r;i++) {`

` if (i != q) {`

` if (i != k) {`

` for (j=0;j<num;j++) {`

` x += formatconstr[i][j] * SOLL[j];`

` if (1 - x < 1e-10) {`

` LF[i]=i;`

` p++;`

` }`

` }`

` x=0;`

` }`

` }`

` }`

` `

` for(g=0;g<p;g++) {`

` q=LF[g];`

` matrix_inverse(formatconstr, q, k);`

` }`

` }`

` if (y == 1) {//case when the angle for the k-th constraint is null`

` matrix_inverse(formatconstr, q, k); //the following provide the inverse of a 2x2 matrix and output solution `

` x=0;`

` p=0;`

` for(i=0;i<r;i++) {`

` if (i != q) {`

` if (i != k) {`

` for (j=0;j<num;j++) {`

` x += formatconstr[i][j] * SOLL[j];`

` if (1 - x < 1e-10) {`

` LF[i]=i;`

` p++;`

` }`

` }`

` x=0;`

` }`

` }`

` }`

` `

` for(g=0;g<p;g++) {`

` q=LF[g];`

` `

` matrix_inverse(formatconstr, q, k);`

` }`

` `

` s++;`

` if (s < num) {`

` goto repeat;`

` }`

` } `

` `

` return 0;`

` //This is the end of the project`

`}`

`void cosine_and_angle(long double array1[8], long double array2[8], int upper_bound)`

`{`

` for(i=0; i < upper_bound ;i++){`

` //compute cosine value of the triangle formed between the variable coefficients, a given constraint, and the origin`

` L8[i] = cosine(array1[i], array2[i], z); `

` `

` //multiply distance, which is measured between the variable coefficients and a given constraints, and the cosine value; this provides distance between the two points if they were aligned between the variable coefficients and the origin; this distance will be referenced as gamma`

` L9[i]=L8[i]*array2[i];`

` `

` //the angle for the cosine`

` L8[i] = acos(L8[i]);`

` }`

`}`

`void min(long double array3[8], int upper_bound)`

`{`

` //x will be assigned the minimum value`

` x = array3[0];`

` for(i=0 ; i < upper_bound ; i++) {`

` if (array3[i] < x) {`

` x = array3[i];`

` }`

` } `

`}`

`void matrix_inverse(long double array4[2][2], int amount6, int amount7) // inverse of 2 x 2 matrix and output of solution`

`{`

` x=0;`

` b=amount6;`

` a=amount7;`

` `

` // matrix containing two constraints meant to be used for obtaining the quantity for each variable`

` `

` //matrix inverse`

` //determinant`

` v = array4[a][0] * array4[b][1] - array4[a][1] * array4[b][0];`

` `

` epi[0][0] = array4[b][1] / v; `

` epi[1][1] = array4[a][0] / v; `

` epi[0][1] = - array4[a][1] / v; `

` epi[1][0] = - array4[b][0] / v; `

` `

` //output of solution`

` for(i=0;i<num;i++){`

` for(j=0;j<num;j++){`

` x += epi[i][j];`

` }`

` SOLL[i]=x;`

` x=0;`

` }`

` sol=0;`

` for(i=0;i<num;i++){`

` sol += N[i] * SOLL[i];`

` }`

` `

` //optimal quantities for each variable are`

` cout<< "\noptimal quantities for each decision variable are respectively "<<SOLL[0]<<" "<< SOLL[1]<< "\n";`

` //optimal solution is `

` cout<< "optimal solution is " <<sol << "\n";`

` `

` //set of optimal constraints which can be combined to determine the optimal quantity for decision variables`

` cout<< "\nthis set of constraints yields the optimal quantity for the decision variables \n";`

` `

` b = LE[b]; //warning: convert double to int`

` a = LE[a]; //warning: convert double to int`

` `

` cout<< "this k-th constraint is pre-requisit for any combination with the set \n["; `

` for(j=0;j<num;j++){`

` cout<<constraints[a][j]<<" ";`

` }`

` cout<<" <= "<<limit[a]<<"\n\n";`

` `

` if ( b > t-1 ) { `

` cout<< "all other constraints within the set are listed \n";`

` for(j=0;j<num;j++){`

` cout<<newconstr[b][j]*limit[a]<<" ";`

` } `

` cout<<"] <= "<<limit[a]<<"\n\n";`

` }`

` `

` if ( b <= t-1 ) { `

` cout<< "all other constraints within the set are listed \n";`

` for(j=0;j<num;j++){`

` cout<<constraints[b][j]<<" ";`

` } `

` cout<<"] <= "<<limit[b]<<"\n\n";`

` } `

` `

`}`

`long double cosine(long double amount1, long double amount2, long double amount3) //function used to compute the cosine `

`{`

` return (pow(amount1,2) - pow(amount2,2) - pow(amount3,2)) / (-2 * amount2 * amount3); `

`}`

`//measure distance between constraint coordinates and variable coefficients, by taking the square root for the sum of squares`

`void dist_constr_var (long double array5[8][2], int amount5, int amount4, long double return_array[8], int upper_bound, int increment)`

`{`

` x=0;`

` `

` for(i=0; i < upper_bound ;i++){`

` `

` u = i + increment; `

` `

` for(j=0;j<num;j++){`

` //measure distance between constraint coordinates and variable coefficients, by taking the square root for the sum of squares`

` x += pow((array5[u][j] - amount5*array5[k][j] - amount4*N[j]), 2);`

` } `

` // measure distance between constraint coordinates and variable coefficients`

` return_array[u] = sqrt(x);`

` x = 0; `

` }`

`}`